Smooth isometric immersions of surfaces into four-space
Smooth isometric immersions of surfaces into four-space is the subject of a manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). The question is whether every closed smooth Riemannian surface can be realized smoothly and isometrically in Euclidean four-space; the manuscript attributes it to Gromov's writings, which trace the four-space surface question to Chern. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original result
- Claim. Every closed smooth Riemannian surface admits a smooth isometric immersion into Euclidean four-space, with no orientability or curvature assumption. The manuscript states that this lowers the immersion dimension from five, in Gromov's construction, to four. Source: the manuscript Smooth isometric immersions of closed surfaces into Euclidean four-space, dated 23 September 2026.
- Formalization, as described by the repository. The scope note lean/docs/333.md states that the formalization proves that every closed smooth Riemannian surface admits a smooth isometric immersion into four-space, with no orientability assumption, and that the statement asks for an immersion rather than an embedding.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math at commit adc7f12, tables at repository commit cd0063d, 13 October 2026) |
Review (fidelity of the formal statements) | Rates this family "full": the Lean statement covers every compact Hausdorff second-countable smooth surface without boundary and every smooth Riemannian metric, with no orientability or curvature hypothesis. Its per-challenge table lists no Comparator run for the challenge SurfaceImmersion. |
The review states that its ratings concern whether the Lean states the headline and "does not say the mathematics is correct beyond what Lean checked"; it describes itself as a model-assisted reading without a referee yet. |
2026-10-09 (latest commit 65cbf4c, 12 October 2026) |
KitaKen1/smooth-isometric-embeddings-r4 (GitHub) | Extension (formal) | States a Lean proof that every closed orientable smooth surface with any smooth Riemannian metric admits a smooth isometric embedding into four-space, built on OpenAI's formalization of the immersion result together with formalized parts of a differential-geometry library and of the classification of surfaces. The README gives an AI-generated sketch in which the metric corrections preserve injectivity. | The repository describes the work as a proposed proof and records the author's run of the Lean Comparator. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.